Summary and Conclusions

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Limitations of Diffeomorphic Transformations

Diffeomorphic transformations are valid for registering images collected from the same individual imaged by two different modalities such as MRI and CT, but it is not necessarily valid when registering images before and after surgery. Likewise, a diffeomorphic mapping assumption may be valid for registering MRI data from two different normal individuals if the goal is to match the deep nuclei of the brain, but it may not be valid for the same data sets if the goal is to match the sulcal patterns.

Alternatively, diffeomorphic transformations may be used to identify areas where two image volumes differ topologically by analyzing the properties of the resulting transformation. For example, consider the problem of matching an MRI image with a tumor to one without a tumor. A possibly valid diffeomorphic transformation would be one that registers all of the corresponding brain structures by shrinking the tumor to a small point. Such a transformation would have an unusually small Jacobian which could be used to detect or identify the location of the tumor. Conversely, consider the inverse problem of matching the image without the tumor to the one with the tumor. A valid registration in this case is to register all of the corresponding brain structures by allowing the transformation to ``tear'' (i.e., not be diffeomorphic) at the site of the tumor[29]. Just as valid could be a diffeomorphic transformation that registers all of the corresponding brain structures by allowing the transformation to stretch at the site of the tumor.

As in the previous examples, we assume that a valid transformation is diffeomorphic everywhere except possibly in regions where the source and target images differ topologically, e.g., in the neighborhood of the tumor. These ideas can be extended to non-diffeomorphic mappings by including the proper boundary conditions around regions that differ topologically.

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Xiujuan Geng 2002-07-04